# What are the prominent algorithms for solving systems of linear inequalities?

The Python sympy symbolic library provides solutions to single, linear Diophantine equations in terms of parametric variables. For instance,

{(t_0, -9*t_0 - 5*t_1 + 154, -5*t_0 - 3*t_1 + 77)}


Without worrying about coding details, how can one solve such a system of parametric expressions for values of the original variables constrained to be, for instance, greater than zero.

Are there any opensource products?

• Sage can do this most likely, although integer linear programming is generally NP-hard. – Kirill Oct 26 '16 at 18:39
• Thank you, yes, I understand. I'm asking because I would like to offer an example involving only a small number of variables, for sympy. – Bill Bell Oct 26 '16 at 18:51
• Can you explain what the syntax of the example means? To me, this simply reads like a collection of linear expressions. How does it represent a system of linear inequalities? – Wolfgang Bangerth Oct 26 '16 at 19:51
• @WolfgangBangerth: Say the Diophantine equation were given in terms of x, y and z. Then one could make arbitrary choice of integers t_0 and t_1 and then one tuple of values for x, y and z would be given by that tuple in the set. I would like to be able to solve, for instance, for the system of inequalities where each item in the tuple is set >0. – Bill Bell Oct 26 '16 at 20:18
• So is there only one solution, or is it a whole region in space that satisfies these equations? Are you only interested in one feasible solution, or in characterizing the whole region? – Wolfgang Bangerth Oct 27 '16 at 14:02